Mathematics

Problem

A differentiable function

g

is defined for all real numbers. Selected values of

g(x)

are recorded in the table.

$x$	$10$	$20$	$40$	$80$	$160$
$g(x)$	$5.80$	$5.40$	$5.20$	$5.10$	$5.05$
$x$	$-10$	$-20$	$-40$	$-80$	$-160$
------	-------	-------	-------	-------	--------
$g(x)$	$-6.20$	$-5.60$	$-5.30$	$-5.15$	$-5.075$

Use the numerical information to analyze the long–range behavior of

g

. (a) Describe the trend in the values of

g(x)

x

increases through the positive values in the table toward larger and larger positive numbers. Cite specific numerical evidence. (b) Describe the trend in the values of

g(x)

x

decreases through the negative values in the table toward more negative numbers. Cite specific numerical evidence. (c) Explain what the trends you identified in parts (a) and (b) suggest about the existence and approximate value(s) of any horizontal asymptote(s) of the graph of

g

. Justify your conclusion with reference to the data.

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